For an $N$-step self-avoiding walk on the hypercubic lattice ${\bf Z}^d$,
we prove that the mean-square end-to-end distance is at least
$N^{4/(3d)}$ times a constant.
This implies that the associated critical exponent $\nu$ is
at least $2/(3d)$, assuming that $\nu$ exists.

Inspired by Aldous' conjecture for
the spectral gap of the interchange process and its recent
resolution by Caputo, Liggett, and Richthammer, we define
an associated order $\prec$ on the irreducible representations of $S_n$. Aldous'
conjecture is equivalent to certain representations being comparable
in this order, and hence determining the ``Aldous order'' completely is a
generalized question. We show a few additional entries for this order.

We show that the SchrÃ¶dinger equation is a lift of Newton's third law
of motion $\nabla^\mathcal W_{\dot \mu} \dot \mu = -\nabla^\mathcal W F(\mu)$ on
the space of probability measures, where derivatives are taken
with respect to the Wasserstein Riemannian metric. Here the potential
$\mu \to F(\mu)$ is the sum of the total classical potential energy $\langle V,\mu\rangle$
of the extended system
and its Fisher information
$ \frac {\hbar^2} 8 \int |\nabla \ln \mu |^2
\,d\mu$. The precise relation is established via a well-known
(Madelung) transform which is shown to be a symplectic submersion
of the standard symplectic
structure of complex valued functions into the
canonical symplectic space over the Wasserstein space.
All computations are conducted in the framework of Otto's formal
Riemannian calculus for optimal transportation of probability
measures.

The groups of (linear) similarity and coincidence isometries of
certain modules $\varGamma$ in $d$-dimensional Euclidean space, which
naturally occur in quasicrystallography, are considered. It is shown
that the structure of the factor group of similarity modulo
coincidence isometries is the direct sum of cyclic groups of prime
power orders that divide $d$. In particular, if the dimension $d$ is a
prime number $p$, the factor group is an elementary abelian
$p$-group. This generalizes previous results obtained for lattices to
situations relevant in quasicrystallography.

In Bernoulli site percolation on Penrose tilings there are
two natural definitions of the critical probability.
This paper shows that they are equal on almost all Penrose tilings.
It also shows that for almost all Penrose tilings the number
of infinite clusters is almost surely~0 or~1.
The results generalize to percolation on a large class of aperiodic
tilings in arbitrary dimension, to percolation on ergodic subgraphs
of $\hbox{\Bbbvii Z}^d$, and to other percolation processes, including
Bernoulli bond percolation.

This article discusses our recent proof that above eight dimensions
the scaling limit of sufficiently spread-out lattice trees is the variant
of super-Brownian motion called {\it integrated super-Brownian excursion\/}
($\ISE$), as conjectured by Aldous. The same is true for nearest-neighbour
lattice trees in sufficiently high dimensions. The proof, whose details will
appear elsewhere, uses the lace expansion. Here, a related but simpler
analysis is applied to show that the scaling limit of a mean-field theory
is $\ISE$, in all dimensions. A connection is drawn between $\ISE$ and
certain generating functions and critical exponents, which may be useful
for the study of high-dimensional percolation models at the critical point.